Dicke quantum spin glass of atoms and photons

Abstract

Recent studies of strongly interacting atoms and photons in optical cavities have rekindled interest in the
Dicke model of atomic qubits coupled to discrete photon cavity modes.
We study the multimode Dicke model with variable atom-photon couplings. We argue that
a quantum spin glass phase can appear, with a random linear combination of the cavity modes superradiant.
We compute atomic and photon spectral response functions across this quantum phase transition, both of which
should be accessible in experiments.

pacs:

37.30.+i, 42.50.-p, 05.30.Rt, 75.10.Nr, 11.30.Qc

Introduction.
Ultracold atoms in optical cavities have emerged as attractive new systems for studying
strongly-interacting quantum many body systems. Photon exchange can mediate long-range interactions
between the atomic degrees of freedom, and this opens up rich possibilities for correlated phases.
In the celebrated atomic realizations of the superfluid-insulator quantum phase transition bloch (),
the light field acts in a secular manner, creating a potential which traps the atoms in an optical lattice;
consequently the atom-atom interactions are only on-site, and this limits the range of possible phases.
In contrast, the seminal recent experiments of Baumann et al.baumann10 (); baumann11 (), realizing a supersolid phase,
have long-range interactions mediated by active photon exchange maschler05 ().

Baumann et al. argued that their experiments could be described by the Dicke model, as in the proposal
of Nagy et al.nagy10 ().
The Dicke model couples photons in a single cavity mode uniformly to N atomic two-level
systems (‘qubits’). In the limit N→∞, this model exhibits a phase transition hepp73 (); wang73 (); emary03 (); ye11 ()
to a “superradiant” phase when the atom-photon
coupling is strong enough. In terms of the qubits, the superradiant phase is a ‘ferromagnet’ which spontaneously breaks a
global Ising symmetry, and so we refer to it as FMSR. In the experiments by Baumann et al.,
the superradiance of the cavity photon mode is accompanied by ‘self-organization’ of the atoms into a density wave
pattern domokos02 (); gopa09 (); gopa10 ().

Here we study extensions of the Dicke model to multiple photon cavity modes, and with non-uniform couplings between
the atomic qubits and the photon modes. Spatial mode variations for the single-mode Dicke model were considered in
Ref. larson09, . Multimode Dicke models have been studied earlier hepp73 (); EK (); thompson (); tolkunov07 (),
but were simplified by ignoring
the variations in the atom-photon couplings. We argue here that qualitatively new physics emerges in the multimode case
when the spatial variation is treated seriously.
We show that large variations in the atom-photon couplings can give rise to a quantum spin-glass (QSG) phase.
We will describe quantum-critical dynamics associated with the onset of this spin glass order.

Dimer et al.dimer07 () have discussed an experimental realization
of the Dicke model using internal atomic degrees of freedom, that is, Raman transitions between multiple atomic levels.
We expect that such schemes can be generalized to a multimode Dicke model that respects a global Ising symmetry, which
is then spontaneously broken in the FMSR and QSG phase, respectively.
More specific realizations of the multimode Dicke model were described recently by Gopalakrishnan et al., in
a paper gopa11 () which appeared while our work was being completed.
The same authors had previously outlined how Bose-Einstein condensates in multimode
cavities can lead to frustration and glassy behavior gopa09 (); gopa10 ().
Such experiments on the multimode Dicke model would provide a unique realization
of a quantum spin glass with long-range couplings, and provide a long-awaited testing ground for theories of quantum systems
with strong interactions and disorder. Condensed matter realizations of quantum spin glasses have shorter-range couplings,
and so do not directly map onto the theoretically solvable systems analyzed in the present paper.

Before describing our computations, we point out a key distinction between the transitions involving onset of
FMSR versus QSG order. In the single-mode Dicke model,
all the qubits align
in a common direction near the FMSR phase, and can therefore be described by a collective spin of length N/2 which behaves classically in the limit of large N.
Consequently, the dynamics near the phase transition can be described by classical equations of motion keeling (), and the single-mode Dicke model does not
realize a quantum phase transition in the conventional sense of condensed matter physics.
In contrast, we will argue here that the onset of QSG order in the multimode Dicke model
has non-trivial quantum fluctuations even in the limit of large N, and the critical properties cannot be described by an
effective classical model. Experimental studies are therefore of great interest.

Model. The Hamiltonian of the multimode Dicke model is

H=M∑i=1ωia†iai+Δ4N∑ℓ=1σzℓ+N∑ℓ=1M∑i=1giℓ(ai+a†i)σxℓ.

(1)

This describes N two-level atomic qubits with level splitting Δ/2 and M photon modes with frequencies ωi
coupled by an atom-photon coupling giℓ which depends on the photon (i) and atom (ℓ) number.
a†i, ai are bosonic creation and annihilation operators, respectively, fulfilling canonical commutation relations.
σx,zℓ are spin-1/2 operators with Pauli matrix representation. As explained in detail in Refs. dimer07 (); gopa11 (), the two states of the Ising spin in Eq. (1) map
onto two different stable ground-state sublevels, |1⟩ and |0⟩, of three-level Λ atoms.
|1⟩ and |0⟩ are indirectly coupled through a pair of Raman
transitions to an excited state |e⟩ which are driven by the classical field of a pair of external lasers.
Upon adiabatic elimination of |e⟩, one obtains Eq. (1) with
σzℓ=|1ℓ⟩⟨1ℓ|−|0ℓ⟩⟨0ℓ|
and σxℓ=|1ℓ⟩⟨0ℓ|+|0ℓ⟩⟨1ℓ|.
The parameters ωi, Δ, and giℓ can be controlled
through laser frequencies and intensities. This tunability enables access to the strong-coupling
Dicke regime.
A dispersive shift of the cavity frequencies
∼a†iajσz does not modify our results significantly, and so will be set to zero for simplicity.
A simple choice for a spatially varying atom-photon coupling is
giℓ=gcos(kixℓ) with ki the wavevector of the photon mode, and xℓ the coordinate
of atom ℓ.

In the single-mode, large photon wavelength case, we have M=1, ωi=ω0,
and giℓ=g/√N and the model can be solved exactly
in the N→∞ limit hepp73 (); wang73 (). At zero temperature,
there is a continuous phase transition between a paramagnetic phase (PM) and
a superradiant ferromagnetic phase (FMSR) at
g=gc=√Δω0/8 at which the Ising symmetry
(a,σx)→(−a,−σx),
is spontaneously broken.

For the multimode Dicke model, it is useful to integrate out the photon degrees of freedom in
a path-integral representation. Then the qubits are described by a Hamiltonian similar to the
Ising model in a transverse field,

Heff=Δ4N∑ℓ=1σzℓ−12∑ℓmJℓmσxℓσxm,

(2)

The exchange interactions Jℓm are mediated by the photons and have a frequency dependence associated with the photon
frequencies ωi; thus Eq. (2) is to be understood as an action appearing in an imaginary time path-integral
summing over time-histories of the qubits. The long-range exchanges

Jℓm(Ω)=M∑i=12giℓgimωiΩ2+ω2i,

(3)

depend on Ω, the imaginary frequency of the qubits in the path integral.
Note that although we have formally integrated
out the photons, we demonstrate below that the photon-photon correlation function is directly related to
the atom-atom correlation function as obtained by solving Eq. (2).

If we ignore the frequency dependence in Eq. (3), the Jℓm have a structure similar to the Hopfield
model of associative memory amit85 (), with M ‘patterns’ giℓ. For M small, it is expected that such
a model can have M possible superradiant ground states with FMSR order ⟨σxℓ⟩∝giℓ, i=1…M. In the spin-glass literature, these are the Mattis states which are “good” memories of
the patterns gamit85 ().
The critical properties of the onset of any of these FMSR states should be similar
to those of the single mode Dicke model.

Our interest in the present paper is focussed on larger values of M, where the summation in Eq. (3) can
be viewed as a sum over M random numbers.
Then, by the central limit theorem, the distribution of Jℓm(Ω) is Gaussian.
Alternatively, the randomness of Jℓm(Ω) can be enhanced by
passing the trapping laser beams through diffusers so that atomic positions xℓ are randomly distributed inside
the cavity gopa11 ().
In either case, we assume a Gaussian distribution characterized by its mean and variance

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Jℓm(Ω)

=

J0(Ω)/N

(4)

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯δJℓm(Ω)δJℓ′m′(Ω′)

=

(δℓℓ′δmm′+δmℓ′δℓm′)K(Ω,Ω′)/N,

where the line represents a disorder average, and δJℓm is the variation from the mean value.
We have assumed couplings between different sites are uncorrelated, and this will allow an exact solution in the N→∞ limit,
modulo an innocuous softening of the
fixed length constraint on the Ising variable ye93 (); read95 (). We will allow arbitrary frequency dependencies in J0(Ω)
and K(Ω,Ω′).
The factors of N ensure an interesting N→∞ limit fischer91 ().
Especially for finite M, one could also use the methods of Ref. amit85 () to extend our analysis to models in which the
giℓ rather than the Jℓm(Ω)
are taken as independent random variables. However, as long as the photon modes can be chosen so that the
Jℓm(Ω) vary in sign and magnitude, our analysis
should remain qualitatively correct also for smaller values of M.

Key results.
We will show below that, in the limit of large atom number N, the results depend only upon
J0(Ω=0) and K(Ω,−Ω).
Here, we will display the phase diagram and spectral response functions for the simple
choices J0(0)=2g2/ω0 and K(Ω,−Ω)≡J2(Ω) with

J(Ω)=2t2ω0/(Ω2+ω20).

(5)

In Fig. 1, we depict the ground
state phase diagram; a related phase diagram in a condensed matter context was obtained in
Ref. dali99, .

All phase transitions are continuous and the respective phase boundaries merge in a bicritical point at (t2bc=0.086,g2bc=t2bc).

Figure 2: (Color online) rf spectral response function of the atomic qubits in the FMSR phase for various photon frequencies and t2=0.025, g2=0.2, Δ=1. Ω is a real measurement frequency. The red arrow at Ω=0 illustrates the delta function contribution with weight
q∼ψ2 from Eqs. (12,13). The value of the gap is given above Eq. (13).
For the Dicke model without disorder (t2=0), the spectral function following from
Eq. (6) consists of nothing but two delta functions: the red arrow at Ω=0 and the grey arrow at
Ω=g√2Δ/ω0 (plotted for ω0=1.015).

The intersection of the PM-FMSR phase boundary with the vertical axis at t2=0 corresponds to the phase transition
in the single mode Dicke
model without disorder emary03 (); dimer07 (). In this case, a number of analytical results can be obtained from
Eq. (11), in agreement with the earlier work. The critical atom-photon coupling is g2c=Δω0/8
and the local σxℓ spin susceptibility in the FMSR phase is (for imaginary frequencies)

Qaaℓ(Ω)∣∣t2=0=ΔΩ2+2Δg2/ω0+ψ22πδ(Ω).

(6)

The corresponding radiofrequency (rf) spectral response function of the atomic qubits for real frequencies,
−Im[Qaa(iΩ→Ω+i0+)], is depicted in Fig. 2.
The superradiance is encoded in the zero frequency
delta function contribution, whose weight is proportional to the atomic population inversion ψ.
However, away from the zero frequency delta function, there is a spectral gap,
and the remaining spectral weight is a delta function at frequency √2Δg2/ω0.

The superradiance also appears as a photon condensate ⟨ai⟩=−∑ℓ(giℓ/(2ωi))⟨σxℓ⟩. We have computed the atomic population inversion,
¯¯¯¯¯¯¯¯¯¯⟨σxℓ⟩=ψ, and the Edwards-Anderson order parameter
¯¯¯¯¯¯¯¯¯¯¯¯⟨σxℓ⟩2=qQSG in Eqs. (13,14). Both of these are related
to ¯¯¯¯¯¯¯¯¯⟨ai⟩, but computation of the latter requires more specific knowledge of the giℓ.
For Ω≠0, the photon correlation function follows from Eq. (6)

⟨a†i(Ω)aj(Ω)⟩=[(iΩ−ωi)δij+N∑ℓ=1giℓgjℓQaaℓ(Ω)]−1,

(7)

where the right-hand-side is a matrix inverse, as can be obtained from integrating out the atomic
fields from the path-integral representation of Eq. (1).

Upon introducing small disorder (with t≠0), as long as we remain in the FMSR phase, the zero frequency delta
function and spectral gap survive, although the higher frequency spectral weight changes, as shown in Fig. 2.
This spectral gap is present across the phase transition from the FMSR phase to the PM phase. Thus all the low energy
fluctuations in the critical theory for this transition are restricted to the zero frequency delta function, which can be described in classical theory for
the spins: this is the reason this transition is more properly considered as a classical phase transition.

Figure 3: (Color online) rf spectral response function of the atomic qubits in the QSG phase for various photon frequencies and t2=0.175, g2=0.05, Δ=1.
The red arrow at Ω=0 illustrates the delta function contribution with weight q∼qQSG from
Eqs. (12,14).

For a sufficiently large value of t2, the system undergoes
a quantum phase transition to the QSG ground state.
In contrast to the PM-FMSR transition, at the QSG transition, and in the entire QSG phase, there is spectral
weight at a continuum of frequencies reaching zero
(see Fig. 3). Thus the onset of QSG order from the PM phase is a genuine quantum phase transition, whose
universality class was described in Ref. read95, .

The PM phase is clearly delineated from both, the QSG and the FMSR phases: the PM phase
has a gapped spectral response and no superradiant photon condensates.

We also note that in all phases, while the spectral function has a universal form at low frequencies,
its high frequency behavior is strongly dependent upon the forms of J0(Ω) and J(Ω).
For the forms in Eq. (5), the spectral function is suppressed to zero at
Ω=ω0.

Measuring the spectrum of photons leaving the cavity through its imperfect mirrors at loss rate κ
allows for an in-situ measurement of our phase diagram, Fig. 1.
Our prediction for the spectrum of intra-cavity photons, Eq. (7), can be
related to the extra-cavity photons via the input-output formalism
dimer07 (); gardiner84 (); walls08 (). For this case of a dissipative Dicke model,
we note a similarity of the decay effects to those in theories of metallic spin glasses sro (),
in which the spin qubits are coupled to a “reservoir” of continuum spin excitations near the Fermi surface.
This coupling leads to a damping term
in the dynamics of each spin, but does not significantly modify the spin-spin interactions responsible for the
spin glass phase. Similarly, for the dissipative Dicke model, decay into photons outside the cavity will
introduce various damping terms e.g. a κ|Ω| term in the denominator of
Eq. (3). As in the previous analyses sro (),
we expect that the quantum spin glass transition will survive in the presence of damping, although there will be
some changes to the critical properties follow-up ().

As in other glasses, we expect slow relaxational dynamics, along with memory and aging effects in the QSG phase
which should be observable via local spin addressing protocols and measuring the spin relaxation time
scale gopa11 ().

Conclusion.
Observations of these effects in quantum optic systems would be remarkable. Moreover,
the spin glass physics is driven by long-range interactions, and this makes
the theoretical models analytically tractable.
We therefore have prospects for a quantitative confrontation between theory
and experiment in a glassy regime, something which has eluded other experimental realizations of
spin glasses.

Details of the calculation.
As discussed in Refs. ye93, ; read95, , each Ising qubit, with on-site gap Δ/2, is conveniently represented
by fluctuations of a non-linear oscillator ϕℓ(τ) (τ is imaginary time) which obeys a unit-length constraint.
Their action at temperature T is then

S0[ϕ,λ]=12ΔN∑ℓ=1∫1/T0dτ[(∂τϕℓ)2+iλℓ(ϕ2ℓ−1)]

(8)

where τ is imaginary time, and the λℓ are Lagrange multipliers which impose the constraints.
The only approximation of this paper is to replace the λℓ by their saddle-point value, iλℓ=λ, and
to ignore their fluctuations.
For decoupled oscillators, this saddle-point value is λ=Δ2/4, the ϕ susceptibility is Δ/(Ω2+Δ2/4),
and the resulting gap, Δ/2, has been matched to that of the Ising spin.

The interactions between the qubits are accounted for as before ye93 (): we introduce replicas a=1…n, average over
the Jℓm using Eq. (4), decouple the resulting two-ϕ coupling by Hubbard-Stratonovich transformation using a ferromagnetic order parameter Ψa(Ω), and the four-ϕ coupling by the bilocal field
Qab(Ω1,Ω2)read95 () (the Ω are Matsubara frequencies).
The complete action is

S=∑aS0[ϕa,λa]+T∑a,ΩJ0(Ω)[N2|Ψa(Ω)|2

−Qab(−Ω,−Ω′)N∑ℓ=1ϕaℓ(Ω)ϕbℓ(Ω′)].

(9)

Now we perform the Gaussian integral over the ϕℓ: the resulting action has a prefactor of N, and so can be replaced
by its saddle-point value. By time-translational invariance, the saddle-point values of the fields can only have the
following frequency dependence

Ψa(Ω)

=

(δΩ,0/T)ψ

Qab(Ω,Ω′)

=

(δΩ+Ω′,0/T)[χ(Ω)δab+(δΩ,0/T)q],

(10)

and we take λa=λ.
We have assumed replica symmetry for the Edwards-Anderson order parameter q because our interest will be limited here
to T=0 where there is no replica symmetry breaking read95 ().
Now the values of the ferromagnetic moment ψ, the
spin susceptibility χ(Ω), q, and λ have to be determined
by optimizing the free energy. The latter is obtained by inserting Eq. (10) in Eq. (9);
after taking the replica limit n→0, we have the free energy per site

F=J0(0)ψ22+T4∑ΩK(Ω,−Ω)|χ(Ω)|2+12K(0,0)χ(0)q

+T2∑Ωln((Ω2+λ)Δ−K(Ω,−Ω)χ(Ω))−λ2Δ

−12[K(0,0)q+J20(0)ψ2λ/Δ−K(0,0)χ(0)].

(11)

Note that this free energy depends only upon J0(0) and K(Ω,−Ω), as claimed earlier.
Our results described in Eq. (6) and Figs. 1-3 are derived from a set
of coupled saddle-point equations obtained from varying Eq. (11) with respect to
χ(Ω), q, ψ, and λ for every Ω. Subsequently we let T→0.

For the choices for K(Ω,−Ω) and J0(0) of Eq. (5),
the rf spectral response function of the atomic qubits plotted in figures 2,3
is given by the expression:

−Im[Qaa(iΩ→Ω+i0+)]=

(12)

∣∣ω20−Ω2∣∣√16Δ2t4ω20−(λ−Ω2)2(ω20−Ω2)28Δt4ω20+q2πδ(Ω).

The first term is non-zero only for frequencies Ω so that the expression underneath the square-root is
positive. The value of the Lagrange multiplier in the FMSR is pinned to
λFM=Δ(J0(0)+K(0,0)/J0(0)). The value of the
gap in Fig. 2 is
√12(λFM+ω20−√16Δt2ω0+(λFM−ω20)2).
This expression equates to zero in the gapless QSG phase shown in Fig. 3,
where λQSG=2Δ√K(0,0).
This gap vanishes logarithmically faster than (t2−t2c) when approaching the QSG phase boundary due to the
square-root behavior of the spectral weight ye93 (); miller93 ().

The ferromagnetic moment obtains as

ψ2=J20(0)−K(0,0)J20(0)(1−∫∞−∞dΩ2πχ(Ω)∣∣λ=λFM),

(13)

and ψ vanishes continuously at the FMSR-QSG phase boundary
(at which J0(0)=√K(0,0)) with exponent βFM=0.5. As per the
discussion above Eq. (7), the corresponding photon condensate
⟨ai⟩ vanishes with the same exponent.
Note that the Edwards-Anderson order parameter q is continuous across this transition
and in the QSG phase given by:

(14)

As expected, one obtains numerically βQSG=1.0=2βFM.

Acknowledgments.
We thank A. Amir, J. Bhaseen, T. Esslinger, J. Keeling, B. Lev, M. Punk, J. Ye, P. Zoller, and especially J. Simon for
useful discussions. This research was supported by the DFG under grant Str 1176/1-1,
by the NSF under Grant DMR-1103860, and by a MURI grant from AFOSR.